[Proof of Proposition REF ]
We first prove the stability guarantee. For this, we use the Rènyi-divergence based analysis used in differential privacy literature.
Let \(P\)  and \(Q\)  be probability distributions such that \(P\)  is absolutely continuous with respect to \(Q\)  and have densities \(\phi _P\)  and \(\phi _Q\) , respectively.
For \(\alpha \in (1,\infty )\) , the \(\alpha \)  Rènyi-divergence between \(P\)  and \(Q\)  is defined as follows [1]}:
\(D_\alpha (P\Vert Q) = \frac{1}{\alpha -1} \ln { \int f_P(x)^\alpha f_Q(x)^{1-\alpha } dx}\) 
